SOLUTION: Given P(x)=2x^5 + 7x^4 - 2x^3 - 25x^2 - 12x + 12 2. Using the DeCartes' rule of signs, state how many positive and how many negative real roots are possible for P(x).(Remember

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Question 131244: Given P(x)=2x^5 + 7x^4 - 2x^3 - 25x^2 - 12x + 12

2. Using the DeCartes' rule of signs, state how many positive and how many negative real roots are possible for P(x).(Remember that rational roots are real!)
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
Using Descartes' Rule of Signs, we can find the possible number of positive roots (x-intercepts that are positive) and negative roots (x-intercepts that are negative)

First lets find the number of possible positive real roots:

For , simply count the sign changes

Here is the list of sign changes:

to (positive to negative)
to (negative to positive)

So there are 2 sign changes, this means there are a maximum of 2 positive roots

So there could be 2, or 0 positive real roots

Now to find the number of negative real roots, we need to find

Replace each x with

Simplify

Now let's count the sign changes for

Here is the list of sign changes:

to (negative to positive)
to (positive to negative)
to (negative to positive)

So there are 3 sign changes, this means there are a maximum of 3 negative roots

So there could be 3, or 1 negative real roots